Linear and Exponential Functions · 线性函数与指数函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ | 线性函数 | xiàn xìng hán shù |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | 指数函数 | zhǐ shù hán shù |
| proportional/prəˈpɔːʃənl/ | 成比例 | chéng bǐ lì |
Add versus multiply, again
- Two savings plans: one adds $\$100$ a year, the other grows by $10\%$ a year.
- At first the flat $\$100$ looks better, but the percentage plan quietly pulls ahead.
- These are the two great patterns of change: linear and exponential.
- They are the continuous cousins of arithmetic and geometric sequences.
又是"加"对"乘"
- 两个储蓄计划:一个每年加 100 元,另一个每年增长 10%。
- 起初那个固定 $\$100$ 的看起来更好,但按百分比的那个悄悄地领先了。
- 这就是变化的两大模式:线性和指数。
- 它们是等差数列和等比数列的连续版表亲。
Arithmetic pairs with linear
- An arithmetic sequence adds a constant; a linear function 线性函数 does the same for every input.
- Over each equal step in $x$, a line adds the same amount to $y$ — its constant slope.
- Plot the arithmetic terms and they land right on a straight line.
- Steady addition is the fingerprint of linear change.
等差对应线性
- 等差数列加一个常数;线性函数(linear function)对每个输入都做同样的事。
- 在 $x$ 上每相等的一步,直线给 $y$ 加上相同的量——它恒定的斜率。
- 把等差各项画出来,它们正好落在一条直线上。
- 稳定的加法是线性变化的指纹。
Over equal input intervals, a linear function changes by… · 在相等的输入区间上,线性函数的变化是……
A line has a constant slope, so equal steps in $x$ add the same amount to $y$ — like an arithmetic sequence. · 直线有恒定的斜率,所以 $x$ 上相等的步长给 $y$ 加上相同的量——就像等差数列。
Geometric pairs with exponential
- A geometric sequence multiplies by a constant; an exponential function 指数函数 does the same continuously.
- Over each equal step in $x$, an exponential multiplies $y$ by the same factor.
- So the outputs form a geometric sequence: $a, ab, ab^2, ab^3, \dots$
- Steady multiplication is the fingerprint of exponential change.
等比对应指数
- 等比数列乘以一个常数;指数函数(exponential function)连续地做同样的事。
- 在 $x$ 上每相等的一步,指数函数把 $y$ 乘以相同的因子。
- 所以输出构成一个等比数列:$a, ab, ab^2, ab^3, \dots$
- 稳定的乘法是指数变化的指纹。
Watch an exponential outrun any straight line · 看指数函数超过任何直线
y = a·bˣ
Raise the base b above 1 and the curve multiplies each step, soon leaving any line behind. Set b below 1 to get decay. · 把底数 b 提到 1 以上,曲线每一步都相乘,很快把任何直线甩在后面。把 b 设到 1 以下就得到衰减。
Over equal input intervals, an exponential function changes by… · 在相等的输入区间上,指数函数的变化是……
An exponential multiplies by the same factor each equal step — the continuous version of a geometric sequence. · 指数函数每相等的一步都乘以相同的因子——它是等比数列的连续版本。
Proportional change
- Exponential growth is proportional 成比例 change: each increase is a percentage of the current amount.
- Grow $10\%$ a year and the yearly gain gets bigger as the total gets bigger.
- Linear growth adds the same absolute amount regardless of the current size.
- "Percentage per period" always signals an exponential, never a line.
成比例的变化
- 指数增长是成比例(proportional)的变化:每次增量都是当前量的一个百分比。
- 每年增长 $10\%$,当总量变大时,每年的增量也变大。
- 线性增长则无论当前大小如何,都加上相同的绝对量。
- "每周期百分之几"总是指数的信号,绝不是直线。
Exponential change is ____ change: the increase is a fixed percentage of the current amount. · 指数变化是____变化:增量是当前量的一个固定百分比。
Proportional change means each step scales with the current value — grow 10% of whatever you have now. · 成比例变化意味着每一步都与当前值成正比——增长当前拥有量的 10%。
A population that doubles every year is best modeled by an exponential function. · 每年翻一倍的人口最适合用指数函数来建模。
Doubling is multiplying by 2 each year — a constant factor over equal intervals, the signature of exponential growth. · 翻倍就是每年乘以 2——相等区间上的恒定因子,正是指数增长的标志。
Select all · 所有 true statements. · 选出所有正确的说法。
Adding equal amounts is linear · 线性, not exponential. The other three correctly pair the ideas. · 加上相等的量是线性的,不是指数的。其余三条正确地对应了这些概念。
Spotting which model
- Look at how the data changes over equal steps: constant difference → linear; constant ratio → exponential.
- A table helps: subtract to test for linear, divide to test for exponential.
- Over a long enough range, exponential growth beats any linear function, no matter how steep.
- Match the model to the pattern of change before you fit numbers.
辨认该用哪个模型
- 看数据在相等步长上如何变化:恒定的差 → 线性;恒定的比 → 指数。
- 用一张表帮忙:相减来检验线性,相除来检验指数。
- 在足够长的范围上,指数增长击败任何线性函数,无论它多陡。
- 在拟合数字之前,先让模型与变化模式匹配。
Do not judge by the first few values. Early on a line can sit above an exponential, so a quick glance misleads. Test the pattern of change — same difference or same ratio — not just the current heights.
不要凭头几个值来判断。早期一条直线可能位于指数之上,所以匆匆一瞥会误导人。检验变化模式——相同的差还是相同的比——而不只是当前的高度。
A pond's lily pads: $10, 20, 40, 80, \dots$ each day.
- Each day the count is multiplied by $2$ — a constant ratio.
- Constant ratio → an exponential function, not a line.
- Rule: pads $= 10 \cdot 2^{\,d}$, doubling without limit until the pond fills.
一个池塘里的睡莲:每天 $10, 20, 40, 80, \dots$。
- 每天数量都乘以 $2$——一个恒定的比。
- 恒定的比 → 一个指数函数,而不是直线。
- 规则:睡莲数 $= 10 \cdot 2^{\,d}$,不断翻倍直到铺满池塘。
An arithmetic sequence pairs with a linear function (adds a constant); a geometric sequence pairs with an exponential function (multiplies by a constant). Exponential growth is proportional — a percentage of the current amount — and it eventually overtakes any line.
等差数列对应线性函数(加一个常数);等比数列对应指数函数(乘以一个常数)。指数增长是成比例的——当前量的一个百分比——它最终会超过任何直线。